Klarinet Archive - Posting 000392.txt from 2001/02

From: Tony@-----.uk (Tony Pay)
Subj: Re: [kl] Combination tones
Date: Sun, 11 Feb 2001 16:15:07 -0500

On Sat, 10 Feb 2001 14:31:50 -0800 (PST), to3456@-----.com said:

> I do agree that the ear is non-linear. It is logarithmic. In being
> logarithmic it can receive audio energy over a much wider range than
> if it was linear. However, I am convinced that the ear is not
> non-linear in the way that would generate difference tones. To put it
> in musical terms, if I hear a trombone and a trumpet separately or
> together I can still recognize them as a trombone and a trumpet. There
> is no third tone. I would expect that if our ears were sensitive to
> difference tones we would all have headaches from the noise.

The fact that the non-linearity of the ear doesn't interfere with that
ability is indeed a miracle, and one that isn't fully understood.
(I hasten to point out that my own degree of understanding of it is very
much less than what is currently understood.)

Nevetheless, it *is* a non-linearity of the sort that generates
difference tones, as is unequivocal in the literature. It's a
non-linearity to do with the way in which the vibration of the eardrum
is transferred to the cochlea, via a system of three connected bones.

> I agree with everyone who says beats are not difference tones. They
> are changes in amplitude.

I've been thinking quite hard about why the explanation of difference
tones in terms of 'speeded-up beats' is seductive, yet wrong. As
someone who for many years thought of the matter in this way, I don't
feel superior to anyone who currently believes in it; even though I
argue against it here, because I now see that it's mistaken.

Certainly, the analogy with a card (credit card?) pulled along the teeth
of a comb -- or a similar sort of effect, like a bit of card organized
to hit the spokes of a bicycle wheel -- is part of the seduction.
Occurring slowly, this produces a sequence of clicks; done faster, a
buzz; and finally, even faster, something like a tone.

What could be more natural than to think that the 'wah-wah-wah-wah' of a
beat between two close frequencies would show the same spectrum of
behaviour?

One way to undermine the analogy (which is what I essentially tried to
do in a previous post) is to see that each comb click actually generates
an impulse on the eardrum; but the beats *don't*.

But there's another seduction.

The analogy tends to convince mathematically inclined people, for the
following reason:

A simple harmonic wave of frequency P can be written as:

sin Pt

...where t is time. (This is a simplification, as is what follows, but
the complications don't affect the argument.)

So the sum of two waves of frequencies P and Q can be written as:

(sin Pt) + (sin Qt)

The processing systems of our ears, from the sum of these two waves of
frequencies P and Q -- which is simple addition of the effects on the
eardrum (the essence of interference, by the way) -- can perceive the
frequencies P and Q.

And if there were three or more waves, the sum could be written as:

(sin Pt) + (sin Qt) + (sin Rt) ...and so on...

...and our ears can similarly perceive the frequencies P, Q, R, etc.

Now the interference explanation of difference tones goes:

------------------------------------------------------------------------

BUT LOOK:

sin Pt + sin Qt = 2 [sin (P + Q)t/2] [cos (P - Q)t/2]

...(which is a standard formula in trigonometry).....

...THEREFORE:

...the sum of (sin Pt) and (sin Qt) can be regarded as the combination
of two waves, one of frequency (P+Q)/2, times one of frequency (P-Q)/2.

So the eardrum takes (P-Q)/2 as a modulation, and therefore as a tone
when large (and at frequency P-Q, because we can't distinguish maxima
and minima).

------------------------------------------------------------------------

But notice that in this argument, the 'combination' rule is no longer
SUMMATION, but MULTIPLICATION.

Therefore the argument doesn't go through, even though it seems
plausible.

For example, notice that if there were a third input R, and R happened
to be equal to (P-Q), then the R = (P-Q) effect on the eardrum would
have to be taken into account.

Would this be by addition, multiplication, or what?

Tony
--
_________ Tony Pay
|ony:-) 79 Southmoor Rd Tony@-----.uk
| |ay Oxford OX2 6RE GMN artist: http://www.gmn.com
tel/fax 01865 553339

... If at first you don't succeed, destroy all evidence that you tried.

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